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Derivation of the Feynman Propagator from Chapter 3 of Student Guide to Quantum Field Theory, by Robert D

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Derivation of the Feynman Propagator from Chapter 3 of Student Guide to Quantum Field Theory, by Robert D Version Date: June 26, 2010 Derivation of the Feynman Propagator From Chapter 3 of Student Guide to Quantum Field Theory, by Robert D. Klauber © 3.0 The Scalar Feynman Propagator The Feynman propagator, the mathematical formulation representing a virtual particle, such as the one represented by the wavy line in Fig. 1-1 of Chap. 1, is the toughest thing, in my opinion, to Feynman propagator not learn and feel comfortable with in QFT. If you don’t feel comfortable with it right away, don’t simple to understand worry about it. That is how virtually everyone feels. Over time, it will become more familiar, and if you are lucky and work hard, maybe even easy. Use wholeness chart as I have tried to take the derivation of the propagator one step at a time, and emphasize what each you study the derivation step entails. Wholeness Chart 5-0X (also at Free Fields Wholeness Chart link at www.quantumfieldtheory.info) breaks these steps out clearly, and should be used as an aid when studying the propagator derivation. Propagators: NRQM vs QFT and Real vs Virtual Particles Note that the propagator for real particles, which you may have studied in NRQM, is not the Feynman propagator for same as the Feynman propagator, which is explicitly for virtual particles in QFT. It may be QFT virtual particles is confusing, but the Feynman propagator is often simply called, “the propagator”. You will have to different from propagator get used to discerning the difference from context. for real particles of NRQM & RQM In QFT, as we will see when we study interactions, a propagator for real particles is not generally needed, and we will not derive one here. 3.0.1 The Approach The first part of QFT is a free particle theory (no interactions, as in this chapter and the next We’ll use the Feynman three). After this, interactions are introduced. In the course of deriving the interaction theory, a propagator when we get mathematical relationship arises that is called the Feynman propagator. Physically, it can be to interaction theory visualized as representing a virtual particle that exists fleetingly and carries energy, momentum, and in some cases, charge from one real particle to another. Thus, it is the carrier, or mediator, of force (interaction.) See the virtual photon of Fig. 1-1 in Chap. 1. It will help us pedagogically to derive the Feynman propagator now, rather than when we get to But it’s easier in the long interactions. The derivation of interaction theory is fairly complicated and it will be easier, as we run if we derive it here develop it, if we already know the mathematical relation for the Feynman propagator, rather than diverting our attention for several pages to derive it then. Heuristically, it may help to consider the virtual particle as created at a particular spacetime point and destroyed at a later spacetime point, and this is how Feynman diagrams portray it. From † this (heuristic) perspective the operator field φ (y) can be considered to create a virtual scalar particle at event y (we used the symbol x2 in Fig. 1-1), and the field operator φ (x) destroys that virtual particle at event x (x1 in Fig. 1-1.) The scalar propagator incorporates these two field operators in a sort of “short-hand” way. Note that the above “creation/destruction at a point” perspective can help initially in understanding the derivation of the propagator, but we caution that it will have to be modified and 39 4 0 0. Error! No text of specified style in document. refined. We will save that to the end when, after digesting the derivation to follow, this modification will be easier to understand. We will now derive a relationship for the propagator using the field operators acting on the vacuum, and will later see (Chap. 7? XXX) that this derived relationship arises naturally in the full mathematical development of the interaction theory. 3.0.2 Milestones in the Derivation We start with the coefficient commutation relations and proceed in five distinct steps. The entire Start with the coefficient derivation is for continuous (not discrete) eigenstate solutions of the field equation (Klein-Gordon commutators and follow here), since the propagator represents a virtual particle in the vacuum and the vacuum is not 5 distinct steps confined to a volume V. We represent the scalar Feynman propagator with the symbol ∆F (x − y) . As noted, our starting point is the coefficient commutators for continuous solutions, Use continuous solutions † † a (k) ,a (k′)⁄ = b (k) ,b (k′)⁄ = δ (k − k′) (continuous) , (3-1) to field equation and from there we follow five steps. The first two are purely mathematical, and will serve as background that will help us with the remaining steps. The third step comprises a physical interpretation of the propagator, and the last two steps are mathematical manipulations to get that interpretation into convenient form (the form that is used in QFT interaction analysis.) Step 1: Use (3-1) to find commutation relations for positive and negative frequency solutions Overview of the 5 steps (see (3-37) in Chap. 3), i.e., determine » ± † _ ÿ ± φ (x) ,φ ( y)⁄= i∆ (x − y) , (3-2) where i∆± (x − y) is the symbol representing the two commutators on the LHS of (3-2). i∆+ represents the upper + and – signs on the LHS; i∆– represents the lower – and + signs. ± ∆ is slightly different from, and will be used to find, the Feynman propagator ∆F. Step 2: Express i∆± (x − y) in terms of a contour integral in the complex plane. Step 3: Express the Feynman propagator ∆F as a mathematical representation of a particle or antiparticle created at one point in space and time in the vacuum and destroyed at another place and time. ± Step 4: Use i∆ (x − y) of Step 2 to express ∆F as a contour integral in the complex plane. Step 5: Re-express ∆F as an integral in real (rather than complex) space. This is the form most suitable for analysis. 3.0.3 The Derivation Step 1: Commutation Relations for Positive/Negative Frequency Solutions Define the symbol i∆+ as the commutator of the field type a solutions, i.e., Step 1, first part, + » + †− ÿ + i∆ (x − y) = φ (x) ,φ ( y)⁄, (3-3) find i∆ = commutation relation for type a fields where the solutions used on the RHS are the integral (continuous) form for the Klein-Gordon solutions (see (3-37) in Chap. 3). It is common usage to use a + sign to designate (3-3), rather than the letter a, which would be easier to remember. Just think “a type field” when you see +. Equation (3-3) is thus ikx ik y 1 e− e ′ + »a ,a† ′ ÿ 3 3 ′ i∆ (x − y) = 3 —— (k) (k )⁄ d k d k 2 (2π ) ωkωk′ (3-4) ik y 1 ≈ e ′ ’ ∆ δ − ′ d 3 ′÷e−ikxd 3 , = 3 — ∆— (k k ) k ÷ k 2 (2π ) « ωkωk′ ◊ and hence, Section 3.0 The Scalar Feynman Propagator 41 −ik ( x− y) + 1 e 3 i∆ (x − y) = 3 — d k . (3-5) 2( 2π ) ωk Similarly, where a minus sign stands for b type fields (since they are associated with antiparticles, the minus makes some sense), Step 1, second part, ik y 1 eikxe− ′ – − » − †+ ÿ » † ′ ÿ 3 3 ′ find i∆ = commutation i∆ (x − y) = φ (x) ,φ ( y)⁄= 3 —— b (k) ,b (k )⁄ d k d k 2( 2π ) ωkωk′ relation for type b fields (3-6) ik ( x− y) −1 e 3 = 3 — d k , 2( 2π ) ωk which leads to _ik ( x− y) _ ± 1 e Step 1, final part, i ± x y » ± x , † y ÿ d 3 (3-7) ∆ ( − ) = φ ( ) φ ( )⁄= 3 — k combine above parts into 2 (2π ) ωk one symbol i∆± ± Step 2: Expressing the Two i as Contour Integrals ∆ Consider the complex plane for a function f of the Review of integral in complex variable k0, i.e., f (k0). Here, the symbol k0 is not a pole (poles are usually designated with null the complex plane subscript), but represents a complex number generalization of the zeroth component (the energy) of 4-momentum k. We concern ourselves with the particular case where k0 takes on the real value ωk. Figure 3-3. Contour Integral From complex variable theory, for Real, Positive Frequency 1 f (k0 ) f (ωk ) = dk0 . (3-8) i2π — k C+ 0 −ωk Now, re-express (3-5) as À −iωk (tx −ty ) ¤ Step 2, first part, 1 ikY x−y Œe Œ + ( ) d 3 + i∆ (x − y) = 3 — e à ‹ k , (3-9) express i∆ as a (2π ) Œ 2ωk Œ contour integral %Õ ((&(('› f ω ( k ) where we take the bracketed quantity as equal to f (ωk), and where −ik t −t e 0 ( x y ) f (k0 ) = . (3-10) k0 + ωk We can then use (3-10) in (3-8) to re-express f (ωk) in terms of a contour integral. Using this for the bracket in (3-9), we find (3-9) becomes 4 2 0. Error! No text of specified style in document. À ¤ + 1 ikY(x−y) Œ 1 f (k0 ) Œ 3 i∆ (x − y) = e à dk0 ‹d k 3 — i2π — k − ω (2π ) ÕŒ C+ 0 k ›Œ À −ik0 (tx −ty ) ¤ 1 ikY(x−y) Œ 1 e Œ 3 = e à dk0 ‹d k (3-11) 3 — i2π — k − ω k + ω (2π ) ÕŒ C+ ( 0 k )( 0 k ) ›Œ −ik x− y −i e ( ) = d 4k . 4 — 2 2 (2π ) C+ (k0 ) − (ωk ) where the integral notation now implies integration over four dimensions of the 4-momentum, with the 3-momentum part from – ∞ to +∞ in real space and the energy part a contour integral in complex space.
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